Let $X$ be a connected CW complex. One can ask to what extent $H_\ast(X)$ determines $\pi_1(X)$. For example, it determines its abelianization, because the Hurewicz Theorem implies that $H_1(X)$ is isomorphic to the abelianization of $\pi_1(X)$.
I'm thinking about invariants of 2-knots which can be extracted from have to do with the second homology of (covers of) their complements, and I'm therefore very much interested in the answer to the following question:

What part of the fundamental group is detected by $H_2(X)$?

In particular, is there an obvious map from $H_2(X)$ (or from part of it) into $\pi_1(X)$? Where in the derived series of $\pi_1(X)$ would the image of $H_2(X)$ live?

From the point of view of the Serre spectral sequence for the Postnikov tower it tells you a little bit about how the Postnikov system twists for the $\pi_2 X$ stage over the $\pi_1 X$ stage. I imagine you could clean that up into a very clear statement but off the top of my head I don't know what it is. In particular you'll need more than just $\pi_1 X$ information. If you're only thinking of $\pi_1 X$ as input there is likely very little (if any) information that survives.
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Ryan BudneySep 29 '10 at 2:18

I guess the comments and answers probably say this already but $\pi_1$ tells you something about $H_2$ and not the other way. An example from Hatcher is that if $pi_1$ is $Z x Z$ you know that $H_2$ is non-vanishing. The basic idea is take your space, attach three cells and higher to build a $K(\pi_1)$ so $H_2(X)$ surjects onto $H_2(\pi_1)$
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Daniel PomerleanoSep 29 '10 at 5:38

A bit off the question maybe, but there is a pullback due to Eilenberg-MacLane that describes the second cohomology group in terms of $\pi_1$ and $\pi_2$ (Determination of the second homology and cohomology groups of a space by means of homotopy invariants. Proc. Nat. Acad. Sci. USA 32 (1946), p. 277–280; see also arxiv.org/abs/0911.2864).
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Matthias KünzerSep 30 '10 at 8:30

7 Answers
7

$H_2(X)$ is all about $\pi_1(X)$ and $\pi_2(X)$. If $\pi_2(X)$ is trivial (as for knot complements) then it is a functor of $\pi_1(X)$.

Let $H_n(G)$ be $H_n(BG)$, the homology of the classifying space ($K(G,1)$). If $X$ is path-connected than there is a surjection $H_2(X)\to H_2(\pi_1(X))$ whose kernel is a quotient of $\pi_2(X)$, the cokernel of a map from $H_3(\pi_1(X))$ to the largest quotient of $\pi_2(X)$ on which the canonical action of $\pi_1(X)$ becomes trivial.

This $H_2(G)$ isn't anything like the next piece of the derived series after $H_1(G)=G^{ab}$, though. For example, if $G$ is abelian then $H_2(G)$ is the second exterior power of $H_1(G)$ (EDIT: so it can be nontrivial even though it knows no more than $H_1(G)$ does), while if $H_1(G)$ is trivial $H_2(G)$ is often nontrivial (EDIT: so, even when it does carry some more information than $H_1(G)$, it is not necessarily derived-series information).

The place to look for the rest of the derived series would be homology with nontrivial coefficients, for example homology of covering spaces.

A slight expansion on my comment, sort of complimentary to Tom's response.

In complete generatlity $H_2X$ tells you nothing about $\pi_1 X$.

If $X = A \times B$ with $A$ a $K(\pi,1)$ and $B$ a $K(\pi,2)$, provided $H_2(A)=0$, you have that $H_2 X = H_2 B$.

Since there are lots of $K(\pi,1)$ spaces with $H_2$ trivial, this allows you to construct many spaces with identical fundamental groups yet $H_2$ varies wildly.

You'll want to restrict to fairly particular spaces to avoid this independence.

edit: If you're happy taking covering spaces then $H_2$ (of of an arbitrary cover of $X$) starts to see quite a bit more of $\pi_1 X$. If $\widetilde{X} \to X$ is the universal cover then $H_2 \widetilde{X} \simeq \pi_2 X$ by the Hurewicz theorem. So now Tom's comments apply, giving you a concrete relationship between $H_2 X$, $\pi_1 X$ and $\pi_2 X = H_2 \widetilde{X}$.

For a group $G$, $H_2(G,\mathbb{Z})$ is also called the Schur multiplier of $G$. Among other things, if $G$ is perfect (ie $H_1=0$) then it is a term in the universal central extension $\widehat{G}$for $G$. That is, you have a short exact sequence

$1\to H_2(G,\mathbb{Z})\to\widehat{G}\to G\to 1$ .

(The wikipedia article focuses on the case of $G$ finite, but this works in greater generality.)

This is closely related to the answers of Henry Wilton and Richard Kent. Namely, $G\wedge G$ is isomorphic to the derived subgroup of a covering group $\hat{G}$ of $G$, and it is not difficult to find an isomorphism between $\ker\kappa$ and $(R\cap [F,F])/[F,R]$.

@Primov You might be interested in the bibliography: bangor.ac.uk/~mas010/nonabtens.html The general construction is given by Brown and Loday in a paper on their van Kampen Theorem, and the above was a bibliogrpahy of papers that mention, use or sometimes are related to that work. The condition that the group be perfect is needed for the universal central extension but there are results as Primoz states for the general case. The constructions are very neat and pretty, and are worth playing with. :-)
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Tim PorterSep 29 '10 at 17:21

A tensor product $G \otimes G$ is introduced so that the commutator map determines a morphism $\kappa: G \otimes G \to G$ whose image is the commutator subgroup. It is proved that Ker $\kappa$ is isomorphic to $\pi_3 S K(G,1)$. In fact the 3-type of $SK(G,1)$ is completely determined. Item 114 of the bibliography Tim mentions links to computer calculations of this tensor product.

The nice point is that $\kappa:G \otimes G \to G$ has, like other tensor products in mathematics, a universal property, and this is not shared by the commutator sugroup of $G$!

thinking about invariants of 2-knots ... is there an obvious map from $H_2(X)$ (or from part of it) into $\pi_1(X)$?
Where in the derived series of $\pi_1(X)$ would the image of $H_2(X)$ live?

Have you heard of things like Dwyer's filtration on $H_2$ and Stallings/Cochran-Harvey theorems about the lower central/derived series? I cannot resist quoting the opening sentence of Krushkal's paper:

"The lower central series of the fundamental group of a space $X$ is closely related to the Dwyer’s [D] ﬁltration $\phi_k(X)$ of the second homology $H_2(X;\Bbb Z)$."

The Dwyer subspace $\phi_k(X)\subset H_2(X;\Bbb Z)$ is deﬁned as the kernel of the composition $$H_2(X) \to H_2(\pi_1X) \to H_2(\pi_1X/\gamma_{k-1}\pi_1X).$$ The gamma notation for the lower central series runs $\gamma_1G=G$, and $\gamma_{k+1}G=[G,\gamma_k G]$.

$\phi_k(X)$ coincides with the set of
homology classes represented by maps of closed $k$-gropes into $X$.

Now their $k$-gropes are gropes of class $k$, but you can also do symmetric gropes
of depth $k$, or whatever they are called, to get into the derived series. Which series to choose, and how to make use of it? The lower central series has a long standing record of applications in knot/link theory due in part to Stallings' theorem (1963, also rediscovered by Casson; note Cochran's topological proof):

Let $\phi:A\to B$ be a homomorphism that induces an isomorphism on $H_1(−;\Bbb Z)$ and an epimorphism on $H_2(−;\Bbb Z)$. Then, for each $n$, $\phi$ induces an isomorphism
$A/\gamma_nA\simeq B/\gamma_nB$.

Now if you do want the derived series rather than the lower central series, take a look at the work of Cochran and Harvey who had a series of papers about the analogues of the Stallings and Dwyer theorems for torsion-free derived series. In fact their motivation was also knot theory.

Also Mikhailov has some other generalization of the Dwyer filtration (see also his book with Passi in Springer Lecture Notes in Math) though I'm not sure if this has had any application to knots.

In fact, I think Mikhailov, and possibly Orr and Cochran, also did something about the transfinite Dwyer filtration, which might be not unrelevant to knots and links (at least this is where the transfinite business originated from, in papers by Orr and Levine in the 80s).